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Assignment 4

Find the following:
(a)
int16x^(-3)dx quad(x!=0)
(d)
int2e^(-2x)dx
(b)
int9x^(8)dx
(e)
int(4x)/(x^(2)+1)dx
(c)
int(x^(5)-3x)dx
(f)
int(2ax+b)(ax^(2)+bx)^(7)dx

Edit $\newline$ $3$ $\newline$ Assignment $4$ $\newline$ $1$ . Find the following: $\newline$ (a) $\int 16 x^{-3} d x \quad(x \neq 0)$ $\newline$ (d) $\int 2 e^{-2 x} d x$ $\newline$ (b) $\int 9 x^{8} d x$ $\newline$ (e) $\int \frac{4 x}{x^{2}+1} d x$ $\newline$ (c) $\int\left(x^{5}-3 x\right) d x$ $\newline$ (f) $\int(2 a x+b)\left(a x^{2}+b x\right)^{7} d x$

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Evaluate definite integrals using the chain rule

Full solution

Q. Edit

\newline

3

\newline

Assignment

4

\newline

1

. Find the following:

\newline

(a)

\int 16 x^{-3} d x \quad(x \neq 0)

\newline

(d)

\int 2 e^{-2 x} d x

\newline

(b)

\int 9 x^{8} d x

\newline

(e)

\int \frac{4 x}{x^{2}+1} d x

\newline

(c)

\int\left(x^{5}-3 x\right) d x

\newline

(f)

\int(2 a x+b)\left(a x^{2}+b x\right)^{7} d x

Problem (a): $\newline$ **Problem (a):** $\newline$ Reasoning: Use the power rule for integration, $\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$ for $n \neq -1$ . $\newline$ Calculation: $\int 16x^{-3} \, dx = 16 \int x^{-3} \, dx = 16 \cdot \frac{x^{-3+1}}{-3+1} + C = -8x^{-2} + C$
Problem (b): $\newline$ **Problem (b):** $\newline$ Reasoning: Apply the power rule for integration. $\newline$ Calculation: $\int 9x^8 \, dx = 9 \cdot \frac{x^{8+1}}{8+1} + C = x^9 + C$

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